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Questions and Answers
What does the cosine rule primarily compare?
What does the cosine rule primarily compare?
- The area of the triangle to its angles
- The ratios of angles in a triangle
- The lengths of the sides of a triangle and the cosine of one of its angles (correct)
- The perimeter of a triangle to the lengths of its sides
In the formula $c^2 = a^2 + b^2 - 2ab ext{cos}(C)$, what condition must apply for this formula to be valid?
In the formula $c^2 = a^2 + b^2 - 2ab ext{cos}(C)$, what condition must apply for this formula to be valid?
- At least one angle must be obtuse
- The triangle must be a right triangle
- The triangle can be any type, including non-right angled (correct)
- The sides must be in ascending order
How does the cosine rule simplify for a right triangle?
How does the cosine rule simplify for a right triangle?
- It applies only to triangles with one angle as 45 degrees
- It changes to a formula involving only the angles
- It becomes the equation for the area of the triangle
- It reduces to the Pythagorean theorem (correct)
What is a key application of the cosine rule beyond basic geometry?
What is a key application of the cosine rule beyond basic geometry?
Which equation correctly describes how to find angle C using the sides of a triangle?
Which equation correctly describes how to find angle C using the sides of a triangle?
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Study Notes
Cosine Rule Mathematics
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Definition: The cosine rule relates the lengths of the sides of a triangle to the cosine of one of its angles.
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Formula: For any triangle with sides ( a ), ( b ), and ( c ), and corresponding opposite angles ( A ), ( B ), and ( C ):
- ( c^2 = a^2 + b^2 - 2ab \cos(C) )
- ( a^2 = b^2 + c^2 - 2bc \cos(A) )
- ( b^2 = a^2 + c^2 - 2ac \cos(B) )
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Uses:
- To find a side length when two sides and the included angle are known.
- To find an angle when all three sides are known.
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Application:
- Works for any triangle (not just right triangles).
- Useful in solving problems in physics and engineering involving non-right angled triangles.
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Special Cases:
- Right triangle: If ( C = 90^\circ ), then ( \cos(C) = 0 ), simplifying to ( c^2 = a^2 + b^2 ) (Pythagorean theorem).
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Example Problems:
- Finding an unknown side:
- Given ( a = 5 ), ( b = 7 ), and ( C = 60^\circ ), calculate ( c ):
- ( c^2 = 5^2 + 7^2 - 2 \times 5 \times 7 \times \cos(60^\circ) )
- Given ( a = 5 ), ( b = 7 ), and ( C = 60^\circ ), calculate ( c ):
- Finding an unknown angle:
- Given ( a = 5 ), ( b = 7 ), and ( c = 10 ), calculate ( C ):
- ( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} )
- Given ( a = 5 ), ( b = 7 ), and ( c = 10 ), calculate ( C ):
- Finding an unknown side:
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Key Points:
- Ensure angles are in the correct unit (degrees or radians) as needed.
- The cosine rule is essential for solving non-right angled triangles efficiently.
Cosine Rule Mathematics
- The cosine rule connects the lengths of the sides of a triangle with the cosine of one of its angles.
- The formula includes three variations depending on which side is being calculated:
- ( c^2 = a^2 + b^2 - 2ab \cos(C) )
- ( a^2 = b^2 + c^2 - 2bc \cos(A) )
- ( b^2 = a^2 + c^2 - 2ac \cos(B) )
Uses of the Cosine Rule
- Utilized to find an unknown side when two sides and the included angle are provided.
- Enables calculation of an unknown angle when all three sides are known.
- Applicable to any triangle type, extending beyond just right triangles.
Application and Relevance
- Frequently used in physics and engineering to solve problems involving non-right angled triangles.
- Simplifies calculations significantly when handling triangles of various configurations.
Special Cases
- In right triangles where ( C = 90^\circ ), the formula reduces to the Pythagorean theorem: ( c^2 = a^2 + b^2 ) since ( \cos(90^\circ) = 0 ).
Example Problems
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Unknown side calculation: For given sides ( a = 5 ), ( b = 7 ) and angle ( C = 60^\circ ), use the formula to find ( c ):
- Apply ( c^2 = 5^2 + 7^2 - 2 \times 5 \times 7 \times \cos(60^\circ) ).
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Unknown angle calculation: Given sides ( a = 5 ), ( b = 7 ), and ( c = 10 ), find angle ( C ):
- Use the formula ( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} ).
Key Points to Remember
- Always check that angle measures are correctly specified in either degrees or radians.
- The cosine rule is fundamental for efficiently solving various triangle-related problems, especially in broader applications beyond traditional geometry.
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